Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 781695, 18 pages
doi:10.1155/2012/781695
Research Article
Influence of Secondary Currents on
Solute Dispersion in Curved Open Channels
Myung Eun Lee1 and Gunwoo Kim2
1
Department of Civil and Environmental Engineering, Seoul National University,
Seoul 151-742, Republic of Korea
2
Department of Ocean Civil & Plant Construction Engineering, Mokpo National Maritime University,
Mokpo, Jeollanamdo 530-729, Republic of Korea
Correspondence should be addressed to Gunwoo Kim, gwkim@mmu.ac.kr
Received 25 December 2011; Accepted 17 April 2012
Academic Editor: Shuyu Sun
Copyright q 2012 M. E. Lee and G. Kim. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
The dispersion coefficient tensor including off-diagonal components was introduced in the flow
with secondary currents, which is called skewed shear flow dispersion SSFD coefficient tensor,
in this paper. To observe the detailed effect of cross-dispersion terms in SSFD model on solute
dispersion, mathematical analysis of eigenvalue problem with respect to the equation with SSFD
coefficient tensor was performed. The analysis results show the several differences of SSFD model
compared to CSFD conventional shear flow dispersion model: the oblique direction of principal
dispersion with respect to the streamline, the increase of peak concentration, and the change in the
eccentricity of elliptical tracer cloud. SSFD coefficient tensor in a streamwise curvilinear coordinate
system of curved channel was transformed to those components of fixed Cartesian coordinate
system, and 2D numerical model with finite element method was established in the EulerianCartesian coordinate. Through this process, the transformation equation using the depth-averaged
velocity field was derived. Several numerical tests were performed to assure the results obtained
in the mathematical analysis and to show the applicability of the derived transformation equation
on the flow with continuously changing flow direction.
1. Introduction
The advection and dispersion of passive solutes in open channels—which includes
pollutant transport in artificial canals, natural streams, and rivers—is an important topic
in environmental hydraulics. In open channels, once vertical mixing is completed in the
initial period of solute transport, the vertical shear velocity profile increases the longitudinal
spreading in the streamline direction 1. As a result, in flows where the longitudinal flow is
2
Journal of Applied Mathematics
y
n
vy
s, vs
Streamline
vx
Pollutant cloud
x
Figure 1: Schematic diagram for CSFD model: x, y: coordinate axes of the Eulerian-Cartesian coordinate
system; vx , vy : components of depth-averaged velocities on the x, y axes.
dominant, such as straight open-channel flows, solute spreading is commonly described with
longitudinal shear dispersion and transverse turbulent diffusion:
∂
∂C
∂C
∂C ∂vs C
∂
Dss
εn
,
∂t
∂s
∂s
∂s
∂n
∂n
1.1
where s is the coordinate axis coinciding with the streamline direction; n is local coordinate
axis that is normal to the streamline; C the depth-averaged concentration; vs the longitudinal
depth-averaged velocity; Dss the longitudinal dispersion coefficient; εn the transverse
turbulent diffusion coefficient. In 1.1, the axis of the longitudinal dispersion always
coincides with the streamline of the principal flow. Thus, the distribution of concentration
shows axisymmetry with respect to the s, n axes, as shown in Figure 1.
However, the secondary current around pronounced curvatures in many open
channels introduces a large magnitude of transverse circulation combined with the principal
longitudinal flow. Hence, the solute dispersion by the secondary current cannot be described
by only the dispersion in the longitudinal direction; there is a dispersion effect in the
transverse direction that is much more effective than the transverse turbulent diffusion. A
flow with secondary currents, as that in Figure 2, has a structure with skewed shear profiles
having different velocity profiles in two orthogonal directions.
Fischer 2 proposed that the cross-dispersion terms should be included in the 2D
depth-averaged dispersion equation, to deal with the effect of skewed vertical profiles on the
horizontal dispersion process:
∂
∂
∂C
∂C
∂C
∂C
∂C ∂vs C
Dss
Dsn
Dns
Dnn
,
∂t
∂s
∂s
∂s
∂n
∂n
∂s
∂n
1.2
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3
x
Streamline
Pollutant cloud
n
s
vn (z)
vs (z)
y
Secondary current
z
Figure 2: Schematic diagram for SSFD model: vs , vn : horizontal velocities on the s, n axes; z: coordinate
axis of the vertical direction.
where Dss , Dsn , Dns , and Dnn are components of the full dispersion coefficient tensor and
Dnn denotes the transverse dispersion coefficient. The additional cross-dispersion terms
Dsn ∂C/∂s and Dns ∂C/∂n indicate mass transport in the longitudinal direction caused
by the concentration gradient in the transverse direction and vice versa. These terms rotate a
dispersing tracer cloud away from symmetry on the s and n axes. Once the open-channel flow
with secondary currents is regarded as the skewed shear flow, the off-diagonal components
Dsn and Dns should be clearly considered on the solute dispersion in a curved flow. However,
most studies and pollutant transport models related to the role of secondary currents in the
dispersion process only focused on activation of the transverse dispersion that only increases
Dnn 3–9. Neglecting cross-dispersion terms, solute dispersion on a curved flow with
secondary currents is still described with only the longitudinal and transverse dispersion
coefficients in the widely used 2D environmental mixing models:
∂
∂
∂C
∂C
∂C ∂vs C
Dss
Dnn
.
∂t
∂s
∂s
∂s
∂n
∂n
1.3
Hereafter, we call this kind of model as a conventional shear flow dispersion CSFD model.
In this study, we introduced the full dispersion coefficient tensor to the solute
dispersion model with respect to the stream wise curvilinear frame of reference as described
in 1.2, and we call this the skewed shear flow dispersion SSFD model. This model
describes the skewed shear flow dispersion process in a flow with a secondary current that
induces strong interaction between the longitudinal and transverse dispersions. For detailed
study with respect to the effect of the off-diagonal terms on the passive solute dispersion, the
dispersion equation with the SSFD coefficient tensor was mathematically analyzed by solving
the eigenvalue problem and comparing the results with the CSFD model. The finite element
formulation in fixed Cartesian coordinates was selected for computational modeling to test
the SSFD model in curved open channels. To deal with the governing equation on a fixed
Cartesian coordinate, a transformation equation for SSFD tensor, originally defined in the
stream wise curvilinear coordinate system, was derived using the depth-averaged velocity
field.
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Journal of Applied Mathematics
2. Analysis of SSFD Model
Two-dimensional advection-dispersion equation is simply expressed below in 2.1 with a
substantial derivative in the stream-wise curvilinear frame of reference
dC
∇ · D∇C ∇T D∇C,
dt
∂ ∂
∇
∂s ∂n
T
2.1
.
By the above expression, the full dispersion coefficient tensor D of the SSFD model is
expressed in a 2 × 2 matrix to correspond to 1.2 as follows:
D
Dss Dsn
Dns Dnn
2.2
.
However in the CSFD model represented by 1.3, D in 2.1 is replaced by the diagonal
dispersion coefficient tensor Dc
dC
∇ · Dc ∇C ∇T Dc ∇C,
dt
Dc Dss
0
0
Dnn
.
2.3
In order to analyze the effect of off-diagonal components on the full dispersion coefficient
tensor of the SSFD model, we performed a mathematical analysis to solve eigenvalue
problems related to the dispersion tensor and equation.
Because ∇T D∇ in 2.1 is in quadratic form, we may replace D with the symmetric
matrix D by taking off-diagonal components together in pairs and writing the result as a sum
of two equal terms 10:
⎡
dC
∇T D∇C,
dt
Dss Dsn
⎤
⎦,
D⎣
Dsn Dnn
2.4
where Dsn Dsn Dns /2. Symmetric dispersion coefficient matrices such as D have an
orthonormal basis of eigenvectors. Thus, if we take these as column vectors, we obtain
a matrix X that is orthogonal—so that XT X−1 . According to the theory of orthogonal
eigenvectors of a specified symmetric matrix, a certain diagonal matrix D is obtained by
the following relationship:
D XD X−1 XD XT .
2.5
It should be noted that D is a similar matrix of D by orthogonal transformation: D X−1 DX.
Substituting 2.5 into 2.4 transforms the quadratic form ∇T D∇ to the principal axes form:
dC
∇T XD XT ∇C
∇ · D ∇C x ,y ,
s,n
dt
2.6
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5
where XT ∇|s,n ∇|x ,y ∂/∂x ∂/∂y T denotes the gradient in the coordinates of
the principal axes x , y . As a result, 2.1 is finally transformed to the equation with the
principal dispersion coefficient tensor D in the rotated x , y -coordinate system. Because D
is a diagonal matrix, the symmetric axes of the equiconcentration line should be parallel to the
directions of the x and y axes. Because the directions of the x and y axes are rotated from
the original s and n axes by the amounts of orthogonal transformations, the axisymmetric
concentration should also be rotated from the s and n axes. We estimated the magnitude of
the rotated angle by solving the following eigenvalue problem:
D∇C λi ∇C,
2.7
where λi i 1, 2 is eigenvalue of the corresponding eigenvector in X, and each value can be
derived as follows:
Dss Dnn
±
λ1 , λ2 2
Dss − Dnn
2
2
2
Dsn .
2.8
The obtained eigenvalues in 2.8 are the diagonal components of D :
D λ1 0
0 λ2
2.9
by the calculation of D X−1 DX. Because the principal dispersion coefficients always have
to be positive, the following is the relationship between each component of D:
Dss Dnn > Dsn .
2.10
By 2.10, we can find the eigenvectors corresponding to λ1 and λ2 that are the column vectors
of X, and the directions of eigenvectors whose directions coincide with the directions of the
x and y axes can be derived. The angle of the counterclockwise rotation ψ of the x and y
axes from the s and n axes is obtained by
⎤
2
D −D
−
D
D
⎥
⎢
ss
nn
ss
nn
±
1⎦.
ψ arctan⎣−
2Dsn
2Dsn
⎡
2.11
Therefore, the angles of ψ represent orthogonal rotation of the axes of symmetry: x , y from
the axes of s, n-coordinates.
Another remarkable effect of the off-diagonal terms in the SSFD model is the
difference in the resultant peak concentration. When we present the analytical solution of
the instantaneously dumped point mass with respect to the x , y -coordinates, it is given by
y2
M
x2
−
,
C x ,y ,t exp −
4λ1 t 4λ2 t
4πht λ1 λ2
2.12
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Journal of Applied Mathematics
where M is the
of tracer. With this solution, the equiconcentration curves are
total mass
ellipses with λ1 and λ2 as the lengths of semimajorand minor axes. Using 2.12, the
peak concentration at time t is inversely proportional to λ1 λ2 , which is given by
λ1 λ2 2
Dss Dnn − Dsn ≤
Dss Dnn .
2.13
According to 2.12 and 2.13, the peak concentration with off-diagonal terms
2
M/4πht Dss Dnn − Dsn is always larger than the peak concentration neglecting offdiagonal terms M/4πht Dss Dnn . Thus, if the off-diagonal components in the SSFD
condition are ignored, the dilution degree of the pollutant is likely to be overestimated.
Assuming that the corresponding CSFD and SSFD models have the same diagonal
components for each dispersion coefficient tensor, the ratio of the overestimated dilution
degree due to neglect of the off-diagonal components by applying the CSFD model is
2
M/ 4πht Dss Dnn
Cp Dc , t
Dss Dnn − Dsn
< 1,
2
Dss Dnn
Cp D, t
M/ 4πht Dss Dnn − Dsn
2.14
where Cp is the peak concentration. Although 2.14 is valid only in the case of instantaneous
release of mass at t 0 in a uniform flow field, the above analytical analysis provides the
knowledge of possible overestimation in dilution degree when the CSFD model is wrongly
applied to a skewed shear flow field of secondary currents.
The final characteristic of the SSFD model compared to the CSFD model is the change
in the eccentricity of ellipses. The dispersive scale in the direction of the symmetric axes of
concentration depends on the pair of principal dispersion coefficients, which are λ1 , λ2 for
the SSFD model and Dss , Dnn for CSFD. Therefore, the comparison between the magnitudes
of the principal dispersion coefficients accounts for the difference in the shapes of the ellipses
by D and Dc . The ellipses of the concentration in the SSFD model have larger eccentricity
than that of the corresponding CSFD model by the relation:
Dss Dnn
max{λ1 , λ2 } 2
Dss Dnn
−
min{λ1 , λ2 } 2
Dss − Dnn
2
Dss − Dnn
2
2
2
Dsn
2
Dsn
2
≥ max{Dss , Dnn },
2.15
≤ min{Dss , Dnn }.
Including 2.15, three characteristics of the SSFD model that contrast with the CSFD
model have been pointed out by the eigenvalue problem solved in this section; the rotation of
the major dispersion axis about the streamline, the larger peak concentration, and the larger
eccentricity of the elliptical concentration. These results show that the application of the CSFD
model to flows with a secondary current is not accurate because skewed vertical profiles
clearly exist in the secondary current combined with the principal flow along the curves.
The oblique direction of the principal dispersion with respect to the streamline and other
characteristics related to the peak concentration and shape of the equiconcentration curves
Journal of Applied Mathematics
7
were demonstrated in a numerical experiment that used the computational model established
in the Eulerian coordinate system, as presented in the next section.
3. Coordinate Transformation
To deal with diverse flow directions in natural streams and rivers with irregular boundaries,
conventional river hydrodynamics and mass transport models are usually established in
a fixed Eulerian coordinate system, where implementing a horizontal unstructured grid is
convenient. In computational models established for such curved channels with continuously
changing flow directions, the principal direction of anisotropic dispersion is usually not
parallel to the axes of the Cartesian coordinates. Therefore, in commonly used CSFD models,
components of Dc constantly defined in a stream-wise curvilinear frame of reference s, n
are transformed into nodal dispersion parameters with respect to the Eulerian-Cartesian
coordinates 11–15. The dispersion coefficient tensor Dc of s, n-coordinates is related to
nodal parameters in x, y-coordinates through the Jacobian matrix expressed with a nodal
velocity vector.
⎤
⎡ ∂x ∂x ⎤ ⎡ v
vy
x
−
⎢ ∂s ∂n ⎥ ⎢
|vs | |vs | ⎥
⎥
⎥⎢
J⎢
⎥,
⎢
⎣ ∂y ∂y ⎦ ⎣ vy
vx ⎦
∂s ∂n
|vs | |vs |
3.1
where vx , vy are depth-averaged velocity components in the x and y directions, respectively.
Because 3.1 is orthogonal matrix, the inverse of J is equal to JT , which results in ∇|s,n JT ∇|x,y . Thus, both 2.1 and 2.3 can then be transformed into equations with respect to
fixed Cartesian coordinate system; by applying ∇|s,n JT ∇|x,y to 2.3,
dC T T T J ∇ Dc J ∇ C ∇T JDc JT ∇C.
dt
3.2
Dc is transformed into the nodal dispersion coefficients with respect to Eulerian-Cartesian
coordinates: JDc JT , where each component of JDc JT can be written as
Dxx Dnn Dss − Dnn Dxy Dyx Dss − Dnn vx 2
vs 2
vx vy
Dyy Dnn Dss − Dnn vs 2
vy 2
vs 2
3.3a
,
,
,
3.3b
3.3c
where Dxx , Dxy , Dyx , and Dyy are Cartesian components of the nodal dispersion coefficient
tensor. As we describe for the commonly used CSFD model, 3.3a, 3.3b, and 3.3c are
the conventional way to determine nodal dispersion parameters by the longitudinal and
transverse dispersion coefficients Dss and Dnn constantly specified in a global domain.
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Journal of Applied Mathematics
The full dispersion coefficient tensor introduced by Fischer 2 also include Dns and
Dsn as cross-dispersion coefficients that can be specified by physical considerations as well as
Dss and Dnn :
1
Dss −
h
Dsn
Dns
Dnn
1
−
h
1
−
h
1
−
h
h
0
h
0
h
0
h
0
vs
vs
vn
vn
z
0
z
0
z
0
z
0
1
εz
1
εz
1
εz
1
εz
z
0
z
0
z
0
z
0
vs dz dz dz,
vn dz dz dz,
3.4
vs
dz dz dz,
vn dz dz dz,
where h is flow depth; ε is the vertical turbulent diffusion coefficient; vs , vn the vertical
deviations of the point velocities with respect to depth-averaged velocities in the s and n
directions, respectively. Because the tensor dispersion coefficients proposed by Fischer 2 is
expressed in fixed coordinates, the same transformation as did in 3.3a, 3.3b, and 3.3c
is needed for components of SSFD tensor to be applied in a flow where the flow direction
continuously changes as in a curved stream: the components of JDJT in SSFD model, which
is analogous to JDc JT of CSFD model, are derived as follows:
Dxx Dss
vx 2
vs 2
− Dsn Dns Dxy Dss − Dnn Dyx Dss − Dnn Dyy Dss
vy 2
vs 2
vx vy
vs 2
vx vy
vs 2
vx vy
vs 2
Dsn
− Dsn
Dsn Dns vx 2
vs 2
vy 2
vs 2
vx vy
vs 2
Dnn
− Dns
Dns
Dnn
vy 2
vs 2
vy 2
vs 2
vx 2
vs 2
vx 2
vs 2
,
,
3.5
,
.
Obviously, 3.5 reduces to 3.3a, 3.3b, and 3.3c if Dsn , Dns are assumed to be zero.
Through 3.5, the coefficients of the SSFD tensor in s, n-coordinates are transformed into
nodal dispersion coefficients in Eulerian x, y-coordinates Figure 2.
Using the transformed component of the nodal CSFD and SSFD coefficient tensor with
respect to the global coordinate system, we present expanded the Cartesian forms of 2.1 and
2.3 as follows:
∂
∂
∂C
∂C
∂C
∂C
∂C ∂vx C ∂ vy C
Dxx
Dxy
Dyx
Dyy
.
∂t
∂x
∂y
∂x
∂x
∂y
∂y
∂x
∂y
3.6
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9
For complex geometries of curved streams with irregular boundaries, an unstructured grid
with the finite element or finite volume method is more useful than a numerical method
with a structure grid. Therefore, we solve 3.6 by the finite element model established in
the Eulerian coordinate system 16. In this model, the Petrov-Galerkin approximation with
a bilinear shape function is applied for spatial discretization, and the Crank-Nicolson type of
discrete time marching is used for transient term. For each nodal point, the components of
the dispersion tensor are determined by 3.3a, 3.3b, and 3.3c in the CSFD model and by
3.5 in the SSFD model.
4. Test for the Direction of Principal Dispersion
In order to observe the oblique direction of the principal dispersion with respect to the
longitudinal streamline, a solute mixing in uniform oscillatory flow was simulated by the
established CSFD and SSFD models. The direction of the oscillatory flow was varied at θ 0◦ ,
30◦ , and 45◦ counterclockwise with respect to the x-axis, to test the applicability of the derived
3.5 for various flow directions. Each result of the SSFD model was compared with that
of the corresponding CSFD model with the same Dss and Dnn . The velocity of the uniform
oscillatory flow was defined by a cosine curve with respect to time as
vx , vy vs cos φt cos θ, vs cos φt sin θ ,
4.1
where vs 0.25 m/s and φ 2π/12 h. In this flow field, the concentration distribution with
a two-dimensional Gaussian profile was given as the initial concentration at the center of a
20 km × 20 km rectangular computational domain; this is the conventional test condition for
the advection-dispersion problem 17. The initial Gaussian distribution was obtained by the
analytic solution after 1 day of pure diffusion of point mass M 5 × 104 kg/m with constant
isotropic diffusion coefficient k 5 m2 /s:
C x, y, 0 x2 y2 M
exp −
−
4πkτ
4k 4k .
4.2
τ1 day
By 4.2, the initial peak concentration was 9.21 ppm. A rectangular element was chosen
as a finite element grid of the test case due to its simplicity. The grid and time step sizes
were determined as Δx Δy 1 km and Δt 900 sec. The maximum Courant number
was vs Δt/Δx 0.225, which guaranteed a stable solution. The initial concentration was
spread under oscillatory flow conditions over 960 time steps, which covered a period of
6 days. Diagonal components of the SSFD and CSFD coefficient tensors of this case were
arbitrarily determined as Dss 10 and Dnn 1 m2 /s. With these longitudinal and transverse
coefficients, the off-diagonal components of the SSFD coefficient were determined as Dsn Dns 3.125 m2 /s, following the dispersion tensor for the application example in Fischer 2:
⎡
⎤
Uo 2 5Uo Vo
⎥
h2 ⎢
192 ⎥
⎢ 120
D ⎢
⎥,
ε ⎣ 5Uo Vo
Vo 2 ⎦
192
12
4.3
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Journal of Applied Mathematics
where Uo , Vo are magnitude of the longitudinal and transverse velocity deviation defined in
Figure 3. The dispersion coefficient in 4.3 was derived from the approximated mean velocity
profile of a continental shelf, as given in Figure 3. When we assume a skewed shear flow
structure for secondary currents as shown in Figure 3, the intensity of the secondary current
is maximized for the same transverse velocity deviation.
The diagonal dispersion coefficient
2
10.979
0
m /s by 2.8 and 2.9, and the coefficient
tensor with eigenvalues of D is D 0
0.021 2
0
tensor of the corresponding CSFD model is Dc 10
0 1 m /s.
In Figure 3, the concentration distributions of the SSFD model CD, t at t 6 days are
compared to those of the corresponding CSFD model CDc , t. In the results from the CSFD
model, which are given in Figures 4a, 4c, and 4e, the direction of the principal dispersion
coincided with the oscillatory flow direction of each case. However, the differences in the
symmetric axes of the concentration by SSFD model and the corresponding oscillatory flow
direction were found to be ψ 17.4◦ , as shown in Figures 4b, 4d, and 4f. This difference
confirmed the eigenvalue
the
2 results given in the previous section. The directions of
10 analysis
◦
3.125
eigenvectors of D 3.125 1 m /s were computed 2.11 to be oriented toward 17.4 and
−72.6◦ with respect to the s axis.
The results in Figure 4 also show the applicability of 3.5; the coordinate transformation of dispersion tensor by 3.5 successfully introduced the full SSFD coefficient tensor in
the numerical grid of Eulerian coordinates. When we consider solute transport in complex
flow with continuously changing flow direction such as meandering streams, the whole
problem is effectively solved in one domain based on the Eulerian frame of reference using
3.5. As presented in the next section, two flows with secondary currents case studies for
application are considered. Through those examples, other aspects of the SSFD model—the
increase in peak concentration and eccentricity of the tracer ellipse—were examined.
5. Application in Flows with Secondary Currents
In order to investigate the performance of the SSFD model in a flow field with secondary
currents, an example case similar to the classic teacup experiment was considered. First, we
assumed a solid-body rotation of water in a coaxial cylindrical container, as in Figure 5, to
produce a so-called forced vortex. In this kind of fluid rotation, there is a pressure gradient
from the perimeter toward the center. When we stop the rotation of the container abruptly,
this pressure gradient coupled with the slower speed near the bottom boundary layer causes
the secondary flow that makes the boundary layer spiral inward to the axis of circulation.
Except for near the side wall and bottom, the fluid continues to rotate as before. Thus, a
rotating flow field of vortex with a constant angular velocity can be assumed for a while in
the container. We assume that a certain passive solute is dropped as an instantaneous point
source as soon as the container is stopped. For this case, both dispersion models were applied
to observe the difference in modeling results: SSFD model accounted for the effect of the
secondary flow and CSFD neglected the secondary flow effect.
The outer and inner cylinders had radii of 10 and 3 m, respectively, and the container
rotated at an angular speed of ω 2π/3 min. The plan view of the container and grid
for the simulation are presented in Figure 6; the grid sizes in the radial and tangential
directions were 0.5 m and πr/40, respectively, where r is the distance from the axis of
rotation.
simulation
was performed by the SSFD model with an arbitrarily determined
0.01The
−0.002 m2 /s of which the magnitude of the off-diagonal component was smaller
D −0.002
0.001
than those of 4.3. The negative sign in the off-diagonal components of D was determined by
Journal of Applied Mathematics
11
y
Vo
x
Uo
z=h
z
Figure 3: Representation of the shear flow on the continental shelf of the middle Atlantic bight 2.
3.3b and 3.3c and the circulating direction of the secondary current, which was negative
n-direction in the upper part of the flow and positive n-direction in the lower
part.
2The
0
dispersion coefficient tensor of the corresponding CSFD model was Dc 0.01
0 0.001 m /s.
An initial concentration of 100 was dropped at the nodal point x, y 6.5 m, 0.
The simulation results at t 36, 72, . . . , 180 sec for both CSFD and SSFD model are
given in Figure 7. Figure 7a shows that the curvilinear axis of the principal dispersion
direction in CDc , t coincided with the circular streamline of the rotating flow. However,
in Figure 7b, the result of CD, t shows that the direction of the major dispersion axis
spiraled outward to the perimeter of the outer cylinder due to the rotation of the symmetric
axis of the tracer cloud. As pointed out in Section 2, the peak concentration of CD, t was
computed to be larger than that of CDc , t; when the tracer cloud rotated for one complete
round at 180 s, Cp Dc , t and Cp D, t were founded to be 3.305 and 3.851, respectively. Along
with the larger peak concentration, the SSFD model increased the eccentricity of the ellipse
of the tracer cloud. The tracer clouds in Figure 7b are longer and narrower than those in
Figure 7a.
For another example case of flow with secondary currents, the dispersion problem in
a strongly curved channel with secondary flow was solved by the SSFD and CSFD modes.
The famous experiments provided by Rozovski 18 were chosen as an example problem, in
which the velocity field was measured along the bend of a U-shaped laboratory channel with
a rectangular cross-section. The width of the channel was 0.8 m with inner and outer bend
radii of 0.4 and 1.2 m, respectively, and the water flowed on the zero-slope bottom with a
mean velocity of 0.25 m/s for his experimental case Run 8.
The velocity field in the whole domain was reproduced by the commonly used depthaveraged flow analysis model RMA-2 in TABS-MD 19. Figure 8 compares the computed
velocities against the measured data of Rozovski 18 at five cross-sections in the bend and
shows that the computational result described the flow field along the channel bend well.
When water flowed along the bend of the rectangular channel, the usual path of velocity
maxima is located near the inner bend at the entrance to the bend and shift to near the outer
Journal of Applied Mathematics
10
10
8
8
6
6
4
4
2
1.5
0
−2
−4
y (km)
y (km)
12
1
0.5
0.2
2
−2
−4
−6
−6
−8
−8
−10
−10 −8 −6 −4 −2
0
2
4
6
8
−10
−10 −8 −6 −4 −2
10
a CSFD model, t 6 day in 0 degree flow
2
4
6
8
10
b SSFD model, t 6 day in 0 degree flow
10
10
8
8
6
6
4
4
2
2
y (km)
y (km)
0
x (km)
x (km)
0
−2
0
−2
−4
−4
−6
−6
−8
−8
−10
−10 −8 −6 −4 −2
0
2
4
6
8
−10
−10 −8 −6 −4 −2
10
c CSFD model, t 6 day in 30 degree flow
8
6
6
4
4
2
2
y (km)
10
8
0
−2
4
6
8
10
0
−2
−4
−4
−6
−6
−8
−8
0
2
d SSFD model, t 6 day in 30 degree flow
10
−10
−10 −8 −6 −4 −2
0
x (km)
x (km)
y (km)
17.4◦
0
2
4
6
8
x (km)
e CSFD model, t 6 day in 45 degree flow
10
−10
−10 −8 −6 −4 −2
0
2
4
6
8
x (km)
f SSFD model, t 6 day in 45 degree flow
Figure 4: Concentration distribution in uniform oscillatory flow unit: ppm.
10
Journal of Applied Mathematics
13
Axis of rotation
ω
y
x
Figure 5: Rotating water in coaxial cylindrical container.
10
y (m)
5
0
−5
−10
−10
−5
0
5
10
x (m)
Figure 6: Grid system for coaxial cylindrical container.
bank at the exit as shown in the measurement of longitudinal velocity in curved channels 20–
22 as observed in Figure 8. This is due to the increase of the water elevation at the outer bank
and the decrease at the inner bank by centrifugal forces in curved channels 23. The principal
flow following the velocity maxima along the bend is accompanied by the secondary flow
by which the path line is partly downstream and partly across the channel from the outer
bank toward the inner bank at the bottom. To observe the solute spreading around the bend,
the initial concentration of 100 was defined at the centered
point of bend
entrance x, y 0.005
−0.002 m2 /s was arbitrarily
6 m, −5.75 m. A dispersion coefficient
tensor
with
D
−0.002 0.001
2
0
selected for the SSFD model, and Dc 0.005
0 0.001 m /s was used for the corresponding CSFD
model.
Concentration distributions at t 4 and 8 s were presented in Figures 9 and 10,
respectively. According to Figures 9 and 10, advection by a lateral nonuniform flow coupled
with the skewed shear flow dispersion causes the remarkable change in the shape of tracer
cloud compared to the CSFD model: the elliptical curves of the equiconcentration computed
14
Journal of Applied Mathematics
0
1
2
3
4
5
6
7
8
9 10 11 12
t = 36 s
t = 72 s
t = 108 s
t = 180 s
t = 144 s
a CSFD
b SSFD
Figure 7: Concentration distribution in coaxial cylindrical container.
−3.5
−4
y (m)
−4.5
−5
−5.5
−6
−6.5
5
5.5
6
6.5
7
7.5
x (m)
0.2 m/s
Rozovski (1965)
Figure 8: Comparison between computed and measured flow fields in Rozovski’s channel.
by the CSFD model maintained the elongated shape along the streamline until the total mass
exited the bend, whereas those of the SSFD model were much shorter. These results seem to be
opposite to the elongated shape of the tracer cloud by the SSFD model shown in the previous
example of a force vortex. The mechanism for the generation of the clustered concentration
by the SSFD model in this example is explained as follows. When the SSFD coefficient tensor
is applied, the direction of the major dispersion axis becomes oblique clockwise with respect
to the streamlines; this is the typical mechanism for skewed shear flow dispersion as well as
Journal of Applied Mathematics
15
1.8
1.5
1.2
0.9
0.6
0.3
0
a CSFD
b SSFD
Figure 9: Concentration distribution in Rozovski’s channel at t 4 s.
1.8
1.5
1.2
0.9
0.6
0.3
0
a CSFD
b SSFD
Figure 10: Concentration distribution in Rozovski’s channel at t 8 s.
the previous example. However, in contrast to the force vortex, the nonuniform longitudinal
velocities of the channel bend shift each particle in the skewed tracer cloud with different
speeds along each streamline; the particles near the inner bank move faster than the particles
near the outer bank. As a result, the dispersed particles in the tracer cloud skewed about
streamlines are centered into the mean displacement of the moving fluids. In particular, at
t 4 s, the shape of the equiconcentration curve appeared to be rounded and concentrated.
The rotation of the major dispersion axis in the SSFD model and the nonuniform
advection along the curved streamline had a combined effect on the increase in the peak
concentration. The peak concentrations at t 8 s, as given in Figure 10, were found to be
0.60 and 0.78 for the CSFD and SSFD models, respectively. This shows that the application
of the CSFD model in the curved channel with a strong secondary current—instead of the
SSFD model—may underestimate the peak concentration of the transported pollutant from
upstream. Although the secondary current is known to activate transverse dispersion and
16
Journal of Applied Mathematics
increase the dilution effect, the results in this study indicate that the skewed vertical shear
profile of the secondary current may offset the enhancement of dilution caused by the large
transverse dispersion.
6. Conclusion
In this study, it was proposed that the SSFD coefficient tensor should be applied for
2D passive solute transport modeling in the flow with secondary current because of
its vertically skewed shear flow structure. Mathematical analysis of eigenvalue problem
pointed out several significant effects of the off-diagonal terms of dispersion tensor: the
rotation of principal direction of dispersion with respect to the streamline, the increase of
peak concentration, and the change in eccentricity of elliptical concentration. To apply full
dispersion coefficient tensor defined in a stream-wise curvilinear coordinate system to the
numerical model on the Eulerian-Cartesian coordinates, transformation relationship was
derived with given depth-averaged velocity field. With the derived transformation equation,
2D numerical model was established with finite element method on the Eulerian coordinate
system. Numerical tests show that the coordinate transformation relationship derived in this
study successfully introduced the SSFD coefficient tensor in the numerical grid of Eulerian
coordinates. It was also shown that there is a possibility of overestimation in dilution of
pollutant if CSFD model was applied instead of SSFD model in the dispersion process
affected by secondary currents. The conventional 2D solute mixing modules equipped in
the various hydrodynamic modeling packages are expected to predict more reliable mixing
patterns of pollutants by including off-diagonal terms as in SSFD model, when it is applied
to flow field with secondary currents.
Nomenclature
C:
Depth-averaged concentration
Peak concentration
Cp :
Dss , Dsn , Dns , and Dnn : Components of the full dispersion coefficient tensor
defined in stream-wise curvilinear coordinates
Dsn :
Mean value of Dsn and Dns
Dxx , Dxy , Dyx , and Dyy : Components of the nodal dispersion coefficient tensor in
the Eulerian-Cartesian coordinates
D:
Matrix notation of SSFD coefficient tensor in curvilinear
coordinates
Matrix notation of CSFD coefficient tensor in curvilinear
Dc :
coordinates
D:
Symmetric version of SSFD coefficient tensor, which
takes Dsn as off-diagonal entries
Principal dispersion coefficient tensor, which takes
D :
eigenvalues λ1 , λ2 as entries
h:
Flow depth
J:
Jacobian matrix for coordinate transformation
k:
Isotropic diffusion coefficient
M:
Total mass of tracer
Journal of Applied Mathematics
n:
s:
t:
Δt:
Uo , Vo :
vs , vn :
vs :
vs , vn :
vx , vy :
x, y:
x , y :
Δx:
Δy:
X:
z:
εn :
ε:
φ:
θ:
λ 1 , λ2 :
ω:
ψ:
17
Axis normal to the streamline in the stream-wise curvilinear coordinate system
Axis along the streamline in the stream-wise curvilinear coordinate system
Time
Discretization size in time
Magnitude of longitudinal and transverse velocity deviations in Figure 3
Horizontal velocities on the s-, n-axis
Longitudinal depth-averaged velocity
Vertical deviations of the point velocities with respect to depth-averaged vs , vn
Ddepth-averaged velocity components in the x and y directions
Axes of the Eulerian-Cartesian coordinate system
Axes with identical directions as the principal axes of D
Discretization size in the x direction
Discretization size in the y direction
Matrix that takes the eigenvectors of D as column vectors
Axis of the vertical direction
Transverse turbulent diffusion coefficient
Vertical turbulent diffusion coefficient
Angular frequency of oscillatory flow
Angle of oscillatory direction with respect to the x-direction
Eigenvalues of D
Angular speed of rotation of coaxial cylindrical container
Angle of counterclockwise rotation from the s axis of the principal dispersion axes.
References
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